function PowerIter
n=500;
A=randn(n,n);
A=(A+A')/2;
for testtime=1:5
y0=randn(n,1);
y0=y0/norm(y0);
[mu,y]=Power(A,y0);
% Store mu and y
mu1(testtime)=mu;
y1(:,testtime)=y;
end
% Display the results
disp('The computed eigenvalues are shown below. You will see they are almost the same.')
disp(mu1)
disp('The inner products of computed eigenvectors are shown below.');
disp('You will see some are close to 1, and some are close to -1.');
disp('This shows that the computed vectors may be in opposite direction');
disp('but both are the same eigenvector.')
disp(y1'*y1)
% Below is the subroutine of power iteration.
function [mu,y]=Power(A,y0)
maxit=10^4;
epsilon=1e-6;
y=y0;
for iter=1:maxit
%%%%%%%
%Insert your lines for Power Iteration
m^(k)=A*y^(k-1)
y^(k)=m^(k)/abs (m^k)_2
U^(k)=(y^(k))'*A*(y^(k))
%%%%%%%
% Check: if ||Ay^k - mu^k y^k ||_2<= epsilon, then break the
% for loop.
% Complete the following line
if
abs (A*y^k - m*u^k y^k)_2 <= epsilon
break;
end
end
end
end
x = [1 2 3 4 5 6 7 8 9 10];
y1 = [.16 .08 .04 .02 .013 .007 .004 .002 .001 .0008 ];
y2 = [.16 .07 .03 .01 .008 .003 .0008 .0003 .00007 .00002 ];
semilogy(x,y1,'-bo;y1;',x,y2,'-kx;y2;');
title('Plot title');
xlabel('X Axis');
ylabel('Y Axis');
print -dpng figure.png
f_cutoff = 2000 f_sample_rate = 10000 nth_order = 2 fnorm = f_cutoff/(f_sample_rate/2); % normalized cut off freq, https://www.exstrom.com/journal/sigproc % Low pass Butterworth filter of order N [b1, a1] = butter(nth_order, fnorm,'low');
%L = imread('images/lena3.jpg');
L = imread('images/Toronto2_F.png');
L=rgb2gray(L);
figure(1);
imshow(L);
b=256; % block size used to look for duplicates
b1=sqrt(b);
Nn = 5; % how many close rows to check
Nd = 15; %Threashold
Nc=26; % Truncate PCA at this length
Nt=26;
% calculate the total size of the image
n = numel(L);
b2 = sqrt(n)-sqrt(b)+1;
% calculate Nb
Nb= power((sqrt(n)-sqrt(b)+1),2);
% the matix of Nc plus the position
M=zeros(Nb, Nc);
% temp index array
Mi = zeros(Nb, 2);
i=1;
disp('Starting PCA');
for r = 1:b2
for c = 1:b2
% Extract each block
B = L(r:r+b1-1,c:c+b1-1);
[pc, latent, explained] = pcacov(cov(double(B)));
%[pc, latent, explained] = princomp(double(B), 'NumComponents', Nc);
Z = pc(1:Nc);
Mi(i,:) = [r c];
M(i,:) = Z;
i = i+1;
end
end
disp('Sorting M -> S');
%Sort M array in lexicographic order -> S
[S, index] = sortrows(M);
P= zeros(1,3);
disp('Finding Duplicates');
for i = 1:Nb
iv = index(i);
xi=mod(iv,b2) + 1;
yi=ceil(iv/b2);
j = i+1;
while j < Nb && abs(i - j) < Nn
jv=index(j);
xj=mod(jv,b2) + 1;
yj=ceil(jv/b2);
z=sqrt(power(xi-xj,2) + power(yi-yj,2));
% only process those whose size is above Nd
if z > Nd
idx = find(P(:,1)== xi & P(:,2)==yi, 1, 'last');
if isempty(idx)==1
P = [P; [xi, yi, 1]];
else
P(idx,3) = P(idx,3) + 1;
end
idx = find(P(:,1)== xi & P(:,2)==yi, 1, 'last');
if isempty(idx)==1
P = [P; [xj, yj, 1]];
else
P(idx,3) = P(idx,3) + 1;
end
end
j = j + 1;
end
end
disp('Sorting findings');
rows = size(P,1);
% sort descending order
P = sortrows(P, -3);
% Mark the found blocks
disp('Creating Image');
idx = 1;
% Create a black image
RI = zeros(sqrt(n), sqrt(n));
while idx < rows && P(idx,3) > 5
x = P(idx,1);
y = P(idx,2);
RI(x,y) = 1;
idx = idx + 1;
end
figure(2);
imshow(RI);
% bai tap 1 a1=10/2\5-3+2*4 b1=3^2/4 c1=3^2^2 d1=2+round(6/9 + 3*2)/2-3 e1=2+floor(6/11)/2-3 f1=2+ceil(-6/9)-3 g1=fix(-4/9)+fix(3*(5/6)) % bai tap 2 a2=mod(36,15) b2=rem(36,15) c2=gcd(36,15) d2=lcm(36,15) % bai tap 3 a3=1 & -1 b3=13 & (-6) c3=0<-20 d3=0<=0.2<=0.4 e3=5>4>3 f3=2>3&1 % bai tap 4 x4=[3 1 5 7 9 2 6] a4=x4(3) b4=x4(1:7) c4=x4(1:end) d4=x4(1:end-1) e4=x4(6:-2:1) f4=x4([1 6 2 1 1]) g4=sum(x4),min(x4),max(x4) % bai tap 5, so sánh từng phần tử của x5 với y5, nếu đúng trả về 1, sai trả về 0 x5=[1 5 2 8 9 0 1] y5=[5 2 2 6 0 0 2] a5=x5>y5 b5=y5<x5 c5= x5==y5 d5 = x5<=y5 e5 = y5>=x5 f5 = x5|y5 g5 = x5&y5 h5 = x5&(-y5) i5=(x5>y5)|(y5<x5) j5=(x5>y5)&(y5<x5) % bài 6 x6=[1 0 2] y6=[0 2 2] x6=y6 b6=x6<y6 c6= x6<y6<x6 d6 = x6<y6<y6 e6 = x6 | (x6) f6 = y6&(y6) x6=y6==x6 % bài 7 x7=1:10 y7=[3 1 5 6 8 2 9 4 7 0] a7=(x7>3)&(x7<8) b7=x7(x7>5) c7=y7(y7<4) d7=x7(x7<2)|(x7>=8) e7=y7(x7<2)|(x7>=8) f7=x7(y7<0) % bài 8 x8=[1 4 8] y8 = [2 1 5] z8=[2 7 9 7;3 1 5 6;8 1 2 5] a8=[x8;y8] b8=z8(:,[1 4]) c8=z8([2 3],[3 1]) d8=z8(:) % bài 13 x13=[1 4 8] y13 = [2 1 5] z13=[3 1 6;5 2 7] a13=x13+y13 b13=z13-3 % bài 14 x14=[2 7 9 7;3 1 5 6;8 1 2 5] a14=x14' b14=sum(x14) c14=sum(x14') d14=sum(x14, 2)
clc
clear
m=0.8;
i=1;
a_m=[18;28];
delta(2,1)=0;
i_matrix(:,1)=0;
a1_matrix(:,1)=0;
a2_matrix(:,1)=0;
while true
jacob=[-sind(a_m(1,1))-sind(a_m(2,1));...
-3*sind(3*a_m(1,1))-3*sind(3*a_m(2,1));...
-5*sind(5*a_m(1,1))-5*sind(5*a_m(2,1));...
-7*sind(7*a_m(1,1))-7*sind(7*a_m(2,1))]
b_func=[2*m;0];
f_func=[cosd(a_m(1,1))+cosd(a_m(2,1));...
3*cosd(3*a_m(1,1))+3*cosd(3*a_m(2,1));..
5*cosd(5*a_m(1,1))+5*cosd(5*a_m(2,1));...
7*cosd(7*a_m(1,1))+7*cosd(7*a_m(2,1))]
delta= jacob/(b_func-f_func);
a_m=acosd(abs(cosd(a_m+delta)));
i_matrix(i,1)=i;
a1_matrix(i,1)=a_m(1,1);
a2_matrix(i,1)=a_m(2,1);
if max(abs(delta))<0.057
break
end
i=i+1;
end
display(sort(a_m));
plot=(i_matrix,a1_matrix,i_matrix,a2_matrix
a=a_m(1,1);
b=a_m(2,1);
t=0.02;
%antistr_simpliromatiki(dna)
ArxikiAllilouxia="AGTAGCAT";
AntistrofiAllilouxia="";
SimbliromatikiAllilouxia="";
for i=1:length(ArxikiAllilouxia)
newStr = substr (ArxikiAllilouxia,i, 1)
fprintf("",newStr);
if (newStr=="A")
SimbliromatikiAllilouxia= strcat(SimbliromatikiAllilouxia, "T")
AntistrofiAllilouxia=strcat("T",AntistrofiAllilouxia)
elseif(newStr=="G")
SimbliromatikiAllilouxia= strcat(SimbliromatikiAllilouxia, "C")
AntistrofiAllilouxia=strcat("C",AntistrofiAllilouxia)
elseif(newStr=="C")
SimbliromatikiAllilouxia= strcat(SimbliromatikiAllilouxia, "G")
AntistrofiAllilouxia=strcat("G",AntistrofiAllilouxia)
elseif(newStr=="T")
SimbliromatikiAllilouxia= strcat(SimbliromatikiAllilouxia, "A")
AntistrofiAllilouxia=strcat("A",AntistrofiAllilouxia)
endif
i=i+1
fprintf("1",SimbliromatikiAllilouxia);
end
L=[20 -1 -1 -1 -1 -1 0 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 -1 -1 0 -1 0 -1 -1 -1 -1 0 0 0 0 ;
-1 8 0 0 0 0 0 -1 0 0 0 0 -1 -1 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
-1 0 2 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
-1 0 0 2 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
-1 0 0 0 2 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
-1 0 0 0 0 7 0 0 -1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 ;
0 0 0 0 0 0 15 -1 -1 0 0 0 0 0 0 0 -1 -1 0 0 -1 0 0 0 -1 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 0 -1 -1 0 0 0 -1 -1 0 0 0 0 0 ;
-1 -1 0 0 0 0 -1 12 -1 0 0 0 -1 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1;
-1 0 0 0 0 -1 -1 -1 21 -1 0 0 0 0 0 0 -1 0 -1 0 -1 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 -1 0 0 0 -1 -1 -1 0 0 0 0 -1 ;
-1 0 0 0 0 0 0 0 -1 5 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 ;
0 0 -1 0 0 0 0 0 0 0 2 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 -1 0 0 0 0 0 0 0 4 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 ;
0 -1 0 0 0 0 0 -1 0 -1 0 0 8 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 -1 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 ;
0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 ;
0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 7 -1 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 -1 0 0 0 -1 -1 -1 -1 0 0 0 -1 0 0 -1 21 0 -1 0 0 -1 0 0 -1 0 -1 -1 0 0 -1 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 -1 0 0 -1 -1 ;
0 -1 0 0 0 0 -1 -1 0 0 0 0 -1 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 ;
0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 -1 0 12 -1 -1 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1;
0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 6 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0;
-1 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 -1 0 7 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 6 -1 0 0 -1 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 37 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 -1 0 -1 0 0 -1 13 0 0 0 0 0 0 0 0 -1 -1 0 0 0 -1 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 ;
0 0 0 0 0 0 -1 0 -1 0 0 0 0 -1 0 0 0 0 -1 0 0 -1 0 -1 0 14 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 -1 0 -1 0 0 0 0 -1 0 0 10 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 ;
0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 -1 0 -1 -1 0 0 0 -1 0 -1 -1 20 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 -1 0 -1 -1 0 -1 0 0 0 0 0 -1 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 7 0 0 -1 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 9 0 0 0 0 -1 0 0 0 0 0 -1 -1 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 10 0 0 -1 0 0 0 -1 -1 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 -1 0 0 0 -1 -1 0 0 10 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 -1 0 0 8 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 -1 0 0 0 -1 -1 0 -1 0 0 14 0 0 0 -1 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 -1 0 -1 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 ;
0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 8 0 0 0 0 0 0 -1 -1 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 -1 0 0 0 -1 0 0 0 7 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 -1 -1 0 0 0 0 -1 0 0 0 0 0 9 0 -1 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 0 0 0 -1 -1 0 -1 -1 0 0 0 0 0 -1 0 16 0 0 0 0 -1 -1 0 0 0 0 0 -1 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 8 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 5 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 -1 0 -1 0 0 0 0 8 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 -1 -1 -1 0 -1 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 15 -1 -1 0 0 0 0 0 0 0 0 -1 0 -1 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 -1 0 0 -1 0 -1 0 0 0 0 0 0 -1 0 0 0 0 -1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 -1 -1 0 0 0 0 0 -1 0 0 0 0 0 0 -1 -1 -1 0 14 0 0 0 0 0 -1 0 -1 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 4 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 4 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 ;
0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 -1 0 0 0 -1 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 -1 -1 -1 0 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 7 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 -1 7 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 -1 0 9 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 0 -1 0 0 0 0 0 0 0 -1 -1 -1 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
-1 -1 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
-1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 ;
-1 0 0 0 0 -1 -1 0 -1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 13 0 0 0 -1 -1 0 0 0 0 0 -1 0 0 0 0 0 ;
-1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 -1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
-1 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 9 0 0 0 0 0 0 0 -1 0 0 0 0 ;
-1 0 0 0 0 0 -1 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 7 -1 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 4 0 0 0 0 0 0 0 0 0 0 ;
-1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 -1 0 0 0 -1 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 7 0 -1 0 -1 0 0 0 -1 ;
-1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 ;
-1 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 13 -1 0 0 0 0 -1 ;
-1 0 0 0 0 -1 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 -1 10 0 0 0 0 0 ;
-1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 -1 0 0 0 12 -1 0 -1 0 ;
0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 6 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 3 0 ;
0 0 0 0 0 0 0 -1 -1 0 0 0 0 -1 0 0 -1 0 -1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 12];
k=eig(L);
for i=1:length(k)
sprintf('%f',k(i,1))
end
function [] = demo()
clc; clear all;
%% Market share draft
Lambdaiwant = 1.501;
% Number of points
npoints = 100;
% Number of muN settings
nmuN = 1;
MSfast = zeros(nmuN,npoints);
MSfastub = zeros(nmuN, npoints);
MSfastlb = zeros(nmuN, npoints);
Lambdavalues = zeros(nmuN, npoints);
muNsettings = [1.1];
settingnumber = 0;
for muN=muNsettings
settingnumber = settingnumber+1
a = 0.5;
low = 0.05;
high = (1 + muN)*0.94;
stepsize = (high - low)/(npoints - 1);
count = 0;
% Original Tmin
Tmin = 10;
for Lambda = low : stepsize : high
count = count + 1
[pis, iMax, jMax, Tmin] = StationaryProbabilities( muN, Lambda, a, 20);
[lambdaslow, lambdafast] = FindArrivalRates( Lambda, iMax, jMax, muN, a, pis );
Lambdavalues(settingnumber,count) = Lambda;
MSfast(settingnumber,count) = lambdafast/Lambda;
end;
end;
end
function [ pis, iMax, jMax, T] = StationaryProbabilities( muN, Lambda, a, Tmin)
%% This script finds the stationary probabilities when both A and N announce. Without loss of generality, let \mu_A = 1 and \mu_N take a higher value
% T is the threshold that we move to attain an epsilon condition
% pis is a matrix of stationary probabilities imax X jmax. i indexes slow server A, j
% indexes fast server N.
% Direction = -1 implies muN is barely less than its value. +1 implies muN is
% barely more than its value.
if(Lambda/(1+muN) > 0.95)
epsilon = 1e-3;
else
epsilon = 1e-4 ;
end;
% What is the minimum I should allow T to be?
% Tmin = 20;
%Lambda = Lambda - 0.0001;
% Initialize
condition = 1;
T = Tmin-1;
% This would go into a loop
while(condition > epsilon)
T = T+1;
iMax = T;
jMax = ceil(T * muN);
% Solve Ax = b
A = zeros((iMax+1) * (jMax+1), (iMax+1) * (jMax+1));
b = zeros((iMax+1) * (jMax+1),1);
% Put in the coefficients for A - one for each state (except 0,0) and one
% for normalization
eqcount = 0;
for i=0:iMax
for j=0:jMax
% Ignore state (0,0)
if(i==0 && j==0)
continue;
end;
% Otherwise
eqcount = eqcount + 1;
b(eqcount) = 0;
% Find the coefficients, which depend on the values of i and
% j
if(i==0)
A(eqcount,Scalarize(i,j,jMax+1)) = Lambda + muN;
A(eqcount,Scalarize(i+1,j,jMax+1)) = -1;
if(j==1)
A(eqcount,Scalarize(0,0,jMax+1)) = -Lambda * a;
end;
if(j<muN*T)
A(eqcount,Scalarize(i,j+1,jMax+1)) = -muN;
end;
continue;
end;
if(j==0)
A(eqcount,Scalarize(i,j,jMax+1)) = Lambda + 1;
A(eqcount,Scalarize(i,j+1,jMax+1)) = -muN;
if(i==1)
A(eqcount,Scalarize(0,0,jMax+1)) = -Lambda *(1-a);
end;
if(i<T)
A(eqcount,Scalarize(i+1,j,jMax+1)) = -1;
end;
continue;
end;
if(i==T && j==muN*T)
A(eqcount,Scalarize(i,j,jMax+1)) = 1 + muN;
A(eqcount,Scalarize(i,j-1,jMax+1)) = -Lambda;
A(eqcount,Scalarize(i-1,j,jMax+1)) = -Lambda;
continue;
end;
if(j==muN*T)
A(eqcount,Scalarize(i,j,jMax+1)) = Lambda + muN + 1;
A(eqcount,Scalarize(i+1,j,jMax+1)) = -1;
A(eqcount,Scalarize(i-1,j,jMax+1)) = -Lambda;
if(j-1 == muN * i)
A(eqcount,Scalarize(i,j-1,jMax+1)) = -Lambda*a;
elseif(j-1 < muN * i)
A(eqcount,Scalarize(i,j-1,jMax+1)) = -Lambda;
end;
%{
if(direction == -1)
if(j-1 <= muN*i)
A(eqcount,Scalarize(i,j-1,jMax+1)) = -Lambda;
end;
elseif(direction == 1)
if(j-1 < muN*i)
A(eqcount,Scalarize(i,j-1,jMax+1)) = -Lambda;
end;
end;
%}
continue;
end;
if(i==T)
A(eqcount,Scalarize(i,j,jMax+1)) = Lambda + muN + 1;
A(eqcount,Scalarize(i,j+1,jMax+1)) = -muN;
A(eqcount,Scalarize(i,j-1,jMax+1)) = -Lambda;
if(j == muN * (i-1))
A(eqcount,Scalarize(i-1,j,jMax+1)) = -Lambda*(1-a);
elseif(j > muN * (i-1))
A(eqcount,Scalarize(i-1,j,jMax+1)) = -Lambda;
end;
%{
if(direction == -1)
if(j >= muN*(i-1))
A(eqcount,Scalarize(i-1,j,jMax+1)) = -Lambda;
end;
elseif(direction == 1)
if(j > muN*(i-1))
A(eqcount,Scalarize(i-1,j,jMax+1)) = -Lambda;
end;
end;
%}
continue;
end;
% Otherwise, it is a generic state (i,j)
A(eqcount,Scalarize(i,j,jMax+1)) = Lambda + muN + 1;
A(eqcount,Scalarize(i+1,j,jMax+1)) = -1;
A(eqcount,Scalarize(i,j+1,jMax+1)) = -muN;
if(j > muN*(i-1))
A(eqcount,Scalarize(i-1,j,jMax+1)) = -Lambda;
elseif (j == muN*(i-1))
A(eqcount,Scalarize(i-1,j,jMax+1)) = -(1-a) * Lambda;
end;
if(j-1 < muN*i)
A(eqcount,Scalarize(i,j-1,jMax+1)) = -Lambda;
elseif(j-1 == muN * i)
A(eqcount,Scalarize(i,j-1,jMax+1)) = -a * Lambda;
end;
%{
if(direction==-1)
if(j >= muN*(i-1))
A(eqcount,Scalarize(i-1,j,jMax+1)) = -Lambda;
end;
if(j-1 < muN*i)
A(eqcount,Scalarize(i,j-1,jMax+1)) = -Lambda;
end;
elseif(direction==1)
if(j > muN*(i-1))
A(eqcount,Scalarize(i-1,j,jMax+1)) = -Lambda;
end;
if(j-1 <= muN*i)
A(eqcount,Scalarize(i,j-1,jMax+1)) = -Lambda;
end;
end;
%}
end;
end;
% Add the normalization equation
eqcount = eqcount + 1;
A(eqcount,:) = 1;
b(eqcount) = 1;
x = A\b;
% Squarify
pis = vec2mat(x,jMax+1);
condition = sum(pis(iMax+1,:))+sum(pis(:,jMax+1));
%condition = pis(iMax+1, jMax+1);
end;
%T
end
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